Holonomic basis
In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold Template:Math is a set of basis vector fields Template:Math defined at every point Template:Math of a region of the manifold as
where Template:Math is the displacement vector between the point Template:Math and a nearby point Template:Math whose coordinate separation from Template:Math is Template:Math along the coordinate curve Template:Math (i.e. the curve on the manifold through Template:Math for which the local coordinate Template:Math varies and all other coordinates are constant).Template:Refn
It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve Template:Math on the manifold defined by Template:Math with the tangent vector Template:Math, where Template:Math, and a function Template:Math defined in a neighbourhood of Template:Math, the variation of Template:Math along Template:Math can be written as
Since we have that Template:Math, the identification is often made between a coordinate basis vector Template:Math and the partial derivative operator Template:Math, under the interpretation of vectors as operators acting on functions.Template:Refn
A local condition for a basis Template:Math to be holonomic is that all mutual Lie derivatives vanish:Template:Refn
A basis that is not holonomic is called an anholonomic,Template:Refn non-holonomic or non-coordinate basis.
Given a metric tensor Template:Math on a manifold Template:Math, it is in general not possible to find a coordinate basis that is orthonormal in any open region Template:Math of Template:Math.Template:Refn An obvious exception is when Template:Math is the real coordinate space Template:Math considered as a manifold with Template:Math being the Euclidean metric Template:Math at every point.